Three manifestly invariant Lagrangians are presented from which the covariant equations of motion for inviscid classical fluids are derived using the least action principle. Invariance and covariance are here defined with respect to synchronous, but otherwise arbitrary, coordinate transformations, i.e. supposing that time intervals are absolute as required by Newtonian mechanics. In the first Lagrangian, the flow is formulated in terms of fluid particles, but conservation of mass and entropy is assumed apriori. In the second Lagrangian, the flow is described by fields, and conservation of mass and entropy is obtained with Lagrange multipliers. The third Lagrangian is also based on a field formulation, but has no Lagrange multipliers and produces all the desired equations of motion, including conservation of mass and entropy. The differences and similarities between these formulations are discussed. Hydrostatic equations are rederived from an asymptotic expansion of the action using the covariant field formulation.